arX

iv:h

ep-p

h/96

0335

3v1

19

Mar

199

6

UCRHEP-T158

March 1996

Efficacious Extra U(1) Factor for

the Supersymmetric Standard Model

E. Keith and Ernest Ma

Department of Physics

University of California

Riverside, California 92521

Abstract

The totality of neutrino-oscillation phenomena appears to require the existence of

a light singlet neutrino. As pointed out recently, this can be naturally accommodated

with a specific extra U(1) factor contained in the superstring-inspired E6 model and

its implied particle spectrum. We analyze this model for other possible consequences.

We discuss specifically the oblique corrections from Z-Z’ mixing, the phenomenology of

the two-Higgs-doublet sector and the associated neutralino sector, as well as possible

scenarios of gauge-coupling unification.

1 Introduction

There are experimental indications at present for three types of neutrino oscillations: solar[1],

atmospheric[2], and laboratory[3]. Each may be explained in terms of two neutrinos differing

in the square of their masses by roughly 10−5 eV2, 10−2 eV2, and 1 eV2 respectively. To

accommodate all three possibilities, it is clear that three neutrinos are not enough. On the

other hand, the invisible width of the Z boson is saturated already with the three known

neutrinos, each transforming as part of a left-handed doublet under the standard electroweak

SU(2)×U(1) gauge group. There is thus no alternative but to assume a light singlet neutrino

which also mixes with the known three doublet neutrinos. As pointed out recently[4], this can

be realized naturally with a specific extra U(1) factor contained in the superstring-inspired

E6 model and its implied particle spectrum.

In Section 2 we map out the essential features of this supersymmetric SU(3)C×SU(2)L×

U(1)Y × U(1)N model. In Section 3 we study the mixing of the standard Z boson with the

Z ′ boson required by the extra U(1)N . We derive the effective contributions of this mixing

to the electroweak oblique parameters ǫ1,2,3 or S, T, U , and show that the U(1)N mass scale

could be a few TeV. In Section 4 we discuss the reduced Higgs potential at the electroweak

scale and show how the two-Higgs-doublet structure of this model differs from that of the

minimal supersymmetric standard model (MSSM). In Section 5 we consider the neutralino

sector and show how the lightest supersymmetric particle (LSP) of this model is constrained

by the Higgs sector. In Section 6 we venture into the realm of gauge-coupling unification

and propose two possible scenarios, each with some additional particles. Finally in Section

7 there are some concluding remarks.

2

2 Description of the Model

The supersymmetric particle content of this model is given by the fundamental 27 represen-

tation of E6. Under SU(3)C × SU(2)L ×U(1)Y ×U(1)N , the individual left-handed fermion

components transform as follows[4].

(u, d) ∼ (3; 2,1

6; 1), uc ∼ (3∗; 1,−2

3; 1), dc ∼ (3∗; 1,

1

3; 2), (1)

(νe, e) ∼ (1; 2,−1

2; 2), ec ∼ (1; 1, 1; 1), N ∼ (1; 1, 0; 0), (2)

(νE , E) ∼ (1; 2,−1

2;−3), (Ec, N c

E) ∼ (1; 2,1

2;−2), (3)

h ∼ (3; 1,−1

3;−2), hc ∼ (3∗; 1,

1

3;−3), S ∼ (1; 1, 0; 5). (4)

As it stands, the allowed cubic terms of the superpotential are uc(uN cE−dEc), dc(uE−dνE),

ec(νeE− eνE), Shhc, S(EEc− νEN

cE), and N(νeN

cE − eEc), as well as hucec, hdcN , and N3.

We now impose a Z2 discrete symmetry where all superfields are odd, except one copy each

of (νE , E), (Ec, N cE), and S, which are even. This gets rid of the cubic terms hucec, hdcN ,

and N3, but allows the quadratic terms hdc, νeNcE − eEc, and N2.

The bosonic components of the even superfields serve as Higgs bosons which break the

gauge symmetry spontaneously. Specifically, 〈S〉 breaks U(1)N and generates mh and mE ;

the electroweak SU(2)L × U(1)Y is then broken by two Higgs doublets as in the MSSM,

with 〈N cE〉 responsible for mu, mD, and m1, and 〈νE〉 for md, me, and m2. The mass matrix

spanning the fermionic components of νe, N , and the odd νE , NcE , and S is then given by

M =

0 mD 0 m3 0

mD mN 0 0 0

0 0 0 mE m1

m3 0 mE 0 m2

0 0 m1 m2 0

, (5)

where the mass term mN is expected to be large because N is trivial under U(1)N and

may thus acquire a large Majorana mass through gravitationally induced nonrenormalizable

3

interactions[5], and m3 comes from the allowed quadratic term νeNcE−eEc. This means that

the usual seesaw mechanism holds for the three doublet neutrinos: mν ∼ m2D/mN , whereas

the two singlet neutrinos have masses mS ∼ 2m1m2/mE and mix with the former through

m3. Note that M is really a 12×12 matrix because there are 3 copies of (νe, N) and 2 copies

of (νE , NcE, S).

3 Z-Z’ Mixing

Let the bosonic components of the even superfields (νE , E), (Ec, N cE), and S be denoted as

follows:

Φ1 ≡

φ01

−φ−1

≡

νE

E

, Φ2 ≡

φ+2

φ02

≡

Ec

N cE

, χ ≡ S. (6)

The part of the Lagrangian containing the interaction of the above Higgs bosons with the

vector gauge bosons Ai (i = 1, 2, 3), B, and Z ′ belonging to the gauge factors SU(2)L, U(1)Y ,

and U(1)N respectively is given by

L = |(∂µ − ig22τiA

µi +

ig12Bµ +

3igN

2√10

Z ′µ)Φ1|2

+ |(∂µ − ig22τiA

µi −

ig12Bµ +

igN√10

Z ′µ)Φ2|2

+ |(∂µ − 5igN

2√10

Z ′µ)χ|2, (7)

where τi are the usual 2× 2 Pauli matrices and the gauge coupling gN has been normalized

to equal g2 in the E6 symmetry limit. Let 〈φ01,2〉 = v1,2 and 〈χ〉 = u, then for

W± =1√2(A1 ∓ iA2), Z =

g2A3 − g1B√

g21 + g22, (8)

we have M2W = (1/2)g22(v

21 + v22), and the mass-squared matrix spanning Z and Z ′ is given

by

M2Z,Z′ =

(1/2)g2Z(v21 + v22) (gNgZ/2

√10)(−3v21 + 2v22)

(gNgZ/2√10)(−3v21 + 2v22) (g2N/20)(25u

2 + 9v21 + 4v22)

, (9)

4

where gZ ≡√

g21 + g22.

Let the mass eigenstates of the Z − Z ′ system be

Z1 = Z cos θ + Z ′ sin θ, Z2 = −Z sin θ + Z ′ cos θ, (10)

then the experimentally observed neutral gauge boson is identified in this model as Z1, with

mass given by

M2Z1

≡ M2Z ≃ 1

2g2Zv

2

[

1−(

sin2 β − 3

5

)2 v2

u2

]

, (11)

where

v2 ≡ v21 + v22 , tan β ≡ v2v1, (12)

and

θ ≃ −√

2

5

gZgN

(

sin2 β − 3

5

)

v2

u2. (13)

The interaction Lagrangian of Z1 with the leptons is now given by

L =

(

1

2gZ cos θ +

gN√10

sin θ

)

νLγµνLZµ1

+

(

(−1

2+ sin2 θW )gZ cos θ +

gN√10

sin θ

)

eLγµeLZµ1

+

(

(sin2 θW )gZ cos θ − gN

2√10

sin θ

)

eRγµeRZµ1 , (14)

where the subscripts L(R) refer to left(right)-handed projections and sin2 θW = g21/g2Z is the

usual electroweak mixing parameter of the standard model. Using the leptonic widths and

the forward-backward asymmetries, the deviations from the standard model are conveniently

parametrized[6]:

ǫ1 =(

sin4 β − 9

25

)

v2

u2= αT, (15)

ǫ2 =(

sin2 β − 3

5

)

v2

u2= − αU

4 sin2 θW, (16)

ǫ3 =2

5

(

1 +1

4 sin2 θW

)(

sin2 β − 3

5

)

v2

u2=

αS

4 sin2 θW, (17)

5

where α is the electromagnetic fine-structure constant. In the above we have also indicated

how Z − Z ′ mixing as measured in the lepton sector would affect the oblique S, T, U pa-

rameters defined originally for the gauge-boson self energies only[7]. The present precision

data from LEP at CERN are consistent with the standard model but the experimental error

bars are of order a few × 10−3[8]. This means that u ∼ TeV is allowed. Note also that the

relative sign of ǫ1,2,3 is necessarily the same in this model.

4 Two-Higgs-Doublet Sector

The Higgs superfields of this model (νE , E), (Ec, N cE), and S are such that the term f(νEN

cE−

EEc)S is the only allowed one in the superpotential. This means that a supersymmetric mass

term for S is not possible and for U(1)N to be spontaneously broken, the supersymmetry

must also be broken. Consider now the Higgs potential. The quartic terms are given by the

sum of

VF = |f |2[(Φ†1Φ2)(Φ

†2Φ1) + (Φ†

1Φ1 + Φ†2Φ2)(χχ)], (18)

and

VD =1

8g22[(Φ

†1Φ1)

2 + (Φ†2Φ2)

2 + 2(Φ†1Φ1)(Φ

†2Φ2)− 4(Φ†

1Φ2)(Φ†2Φ1)]

+1

8g21[(Φ

†1Φ1)

2 + (Φ†2Φ2)

2 − 2(Φ†1Φ1)(Φ

†2Φ2)]

+1

80g2N [9(Φ

†1Φ1)

2 + 4(Φ†2Φ2)

2 + 12(Φ†1Φ1)(Φ

†2Φ2)− 30(Φ†

1Φ1)(χχ)

− 20(Φ†2Φ2)(χχ) + 25(χχ)2]. (19)

The soft terms which also break the supersymmetry are given by

Vsoft = µ21Φ

†1Φ1 + µ2

2Φ†2Φ2 +m2χχ+ fAΦ†

1Φ2χ+ (fA)∗χΦ†2Φ1. (20)

The first stage of symmetry breaking occurs with 〈χ〉 = u. From Vsoft and VD, we see

that u2 = −8m2/5g2N . Consequently,√2Imχ combines with Z ′ to form a massive vector

6

gauge boson and√2Reχ is a massive scalar boson. Both have the same mass:

M2Z′ = m2

χ =5

4g2Nu

2. (21)

The reduced Higgs potential involving only the two doublets is then of the standard form:

V = m21Φ

†1Φ1 +m2

2Φ†2Φ2 +m2

12(Φ†1Φ2 + Φ†

2Φ1)

+1

2λ1(Φ

†1Φ1)

2 +1

2λ2(Φ

†2Φ2)

2 + λ3(Φ†1Φ1)(Φ

†2Φ2) + λ4(Φ

†1Φ2)(Φ

†2Φ1), (22)

where

m21 = µ2

1 −3

8g2Nu

2, m22 = µ2

2 −1

4g2Nu

2, m212 = fAu, (23)

assuming that f and A are real for simplicity. In the above, we have of course also assumed

implicitly that m21, m

22, and m2

12 are all small in magnitude relative to u2. The quartic scalar

couplings λ1,2,3,4 receive contributions not only from the coefficients of the corresponding

terms in VD and VF , but also from the cubic couplings of√2Reχ to the doublets which are

proportional to u, as shown in Fig. 1. As a result[9],

λ1 =1

4(g21 + g22) +

9

40g2N − 8(f 2 − 3g2N/8)

2

5g2N=

1

4(g21 + g22) +

6

5f 2 − 8f 4

5g2N, (24)

λ2 =1

4(g21 + g22) +

1

10g2N − 8(f 2 − g2N/4)

2

5g2N=

1

4(g21 + g22) +

4

5f 2 − 8f 4

5g2N, (25)

λ3 = −1

4g21 +

1

4g22 +

3

20g2N − 8(f 2 − 3g2N/8)(f

2 − g2N/4)

5g2N

= −1

4g21 +

1

4g22 + f 2 − 8f 4

5g2N, (26)

λ4 = −1

2g22 + f 2. (27)

It is obvious from the above that the two-Higgs-doublet sector of this model differs from that

of the minimal supersymmetric standard model (MSSM) and reduces to the latter only in

the limit f = 0. Note that if m212 is of order m

2χ, then it is not consistent to assume that both

Φ1 and Φ2 are light. In that case, only a linear combination of Φ1 and Φ2 may be light and

7

the electroweak Higgs sector reduces to that of just one doublet, as in the minimal standard

model.

Since V of Eq. (22) should be bounded from below, we must have

λ1 > 0, λ2 > 0, λ1λ2 − (λ3 + λ4)2 > 0 if λ3 + λ4 < 0. (28)

Hence f 2 has an upper bound. For g2N = (5/3)g21 which is a very good approximation if

U(1)Y and U(1)N are unified only at a very high energy scale, we find that the ratio f 2/g2Z

has to be less than about 0.35. After electroweak symmetry breaking, the upper bound on

the lighter of the two neutral scalar Higgs bosons is given in general by

(m2h)max = 2v2[λ1 cos

4 β + λ2 sin4 β + 2(λ3 + λ4) sin

2 β cos2 β] + ǫ, (29)

where ǫ comes from radiative corrections, the largest contribution being that of the top

quark:

ǫ ≃ 3g22m4t

8π2M2W

ln

(

1 +m2

m2t

)

, (30)

with m coming from soft supersymmetry breaking. In the present model, this becomes

(m2h)max = 2v2

[

1

4g2Z cos2 2β + f 2

(

3

2+

1

5cos 2β − 1

2cos2 2β

)

− 8f 4

5g2N

]

+ ǫ. (31)

Considered as a function of f 2, the above quantity is maximized at

f 20 =

5g2N16

(

3

2+

1

5cos 2β − 1

2cos2 2β

)

. (32)

Assuming that g2N = (5/3)g21 as before, we find f 20 /g

2Z to be always smaller than the upper

bound we obtained earlier from requiring V > 0. Hence we plot (mh)max in Fig. 2 for f = f0

and f = 0 as functions of cos2 β, as the maximum allowed values of mh in this model and

in the MSSM respectively. It is seen that for mt = 175 GeV and m = 1 TeV, mh may be as

high as 140 GeV in this model, as compared to 128 GeV in the MSSM.

8

For the charged Higgs boson H± = sin βφ±1 − cos βφ±

2 and the pseudoscalar Higgs boson

A =√2(sin βImφ0

1 − cos βImφ02), we now have the sum rule

m2H± = m2

A +M2W − f 2v2, (33)

where m2A = −m2

12/ sinβ cos β. Note that the above equation is common to all extensions[9]

of the MSSM with the term fΦ†1Φ2χ in the superpotential and would serve as an unambiguous

signal of physics beyond the MSSM at the supersymmetry breaking scale.

5 The Neutralino Sector

In the MSSM, there are four neutralinos (two gauge fermions and two Higgs fermions) which

mix in a well-known 4 × 4 mass matrix[10]. Here we have six neutralinos: the gauginos of

U(1)Y and the third component of SU(2)L, the Higgsinos of φ01 and φ0

2, the U(1)N gaugino

and the χ Higgsino. The corresponding mass matrix is then given by

MN =

M1 0 −g1v1/√2 g1v2/

√2 0 0

0 M2 g2v1/√2 −g2v2/

√2 0 0

−g1v1/√2 g2v1/

√2 0 fu −3gNv1/2

√5 fv2

g1v2/√2 −g2v2/

√2 fu 0 −gNv2/

√5 fv1

0 0 −3gNv1/2√5 −gNv2/

√5 M1

√5gNu/2

0 0 fv2 fv1√5gNu/2 0

,

(34)

where M1,2 are allowed U(1) and SU(2) gauge-invariant Majorana mass terms which break

the supersymmetry softly. Note that without the last two rows and columns, the above

matrix does reduce to that of the MSSM if fu is identified with −µ. Recall that if f is very

small, then the two-Higgs-doublet sector of this model is essentially indistinguishable from

that of the MSSM, but now a difference will show up in the neutralino sector unless the µ

parameter of the MSSM accidentally also happens to be very small. In other words, there is

an important correlation between the Higgs sector and the neutralino sector of this model

9

which is not required in the MSSM.

Since gNu cannot be small, the neutralino mass matrix MN reduces to either a 4× 4 or

2×2 matrix, depending on whether fu is small or not. In the former case, it reduces to that

of the MSSM but with the stipulation that the µ parameter must be small, i.e. of order 100

GeV. This means that the two gauginos mix significantly with the two Higgsinos and the

lightest supersymmetric particle (LSP) is likely to have nonnegligible components from all

four states. In the latter case, the effective 2× 2 mass matrix becomes

M′N =

M1 + g21v1v2/fu −g1g2v1v2/fu

−g1g2v1v2/fu M2 + g22v1v2/fu

. (35)

Since v1v2/u is small, the mass eigenstates of M′N are approximately the gauginos B and

A3, with masses M1 and M2 respectively. In supergravity models,

M1 =5g213g22

M2 ≃ 0.5 M2, (36)

hence B would be the LSP.

In the chargino sector, the corresponding mass matrix is

Mχ =

M2 g2v2

g2v1 −fu

. (37)

If fu is small, then both charginos can be of order 100 GeV, but if fu is large (say of

order 1 TeV), then only one may be light and its mass would be M2. In the MSSM, the

superpotential has the allowed term µΦ†1Φ2. Hence there is no understanding as to why µ

should be of order of the supersymmetry breaking scale, and not in fact very much greater.

Here fu is naturally of order of the U(1)N breaking scale, and since the latter cannot be

broken without also breaking supersymmetry, the two scales are necessarily equivalent. This

solves the so-called µ problem of the MSSM.

10

6 Gauge-Coupling Unification

In the MSSM, the three gauge couplings g3, g2, and gY = (5/3)1

2 g1 have been shown to

converge to a single value at around 1016 GeV[11]. In the present model, with particle content

belonging to complete 27 representations of E6 and nothing else, this unification simply does

not occur. This is a general phenomenon of all grand unified models: the experimental values

of the three known gauge couplings at the electroweak energy scale are not compatible with

a single value at some higher scale unless the particle content (excluding the gauge bosons)

has different total contributions to the evolution of each coupling as a function of energy

scale. The evolution equations of αi ≡ g21/4π are generically given to two-loop order by

µ∂αi

∂µ=

1

2π

[

bi +bij4π

αj(µ)

]

α2i (µ), (38)

where µ is the running energy scale and the coefficients bi and bij are determined by the

particle content of the model. To one loop, the above equation is easily solved:

α−1i (M1) = α−1

i (M2)−bi2π

lnM1

M2

. (39)

Below MSUSY, assume the standard model with two Higgs doublets, then

bY =21

5, b2 = −3, b3 = −7. (40)

Above MSUSY in the MSSM,

bY = 3(2) +3

5(4)

(

1

4

)

, b2 = −6 + 3(2) + 2(

1

2

)

, b3 = −9 + 3(2). (41)

Note that in the above, the three supersymmetric families of quarks and leptons contribute

equally to each coupling, whereas the two supersymmetric Higgs doublets do not. The

reason is that the former belong to complete representations of SU(5) but not the latter.

For MSUSY ∼ 104 GeV, the gauge couplings would then unify at MU ∼ 1016 GeV in the

MSSM.

11

In the present model as it is, the one-loop coefficients of Eq. (38) above MSUSY(∼ u) are

bY = 3(3), b2 = −6 + 3(3), b3 = −9 + 3(3), bN = 3(3), (42)

because there are three complete 27 supermultiplets of E6. [Actually N is superheavy but it

transforms trivially under SU(3)C × SU(2)L × U(1)Y × U(1)N .] To achieve gauge-coupling

unification, we must add new particles in a judicious manner. One possibility is to mimic

the MSSM by adding one extra copy of the anomaly-free combination (νe, e) and (Ec, N cE).

Then

∆bY =3

5, ∆b2 = 1, ∆b3 = 0, ∆bN =

2

5. (43)

Since the relative differences of bY , b2, and b3 are now the same as in the MSSM, we have

again unification at MU ∼ 1016 GeV, from which we can predict the value of gN at MSUSY .

We show in Fig. 3 the evolution of α−1i using also the two-loop coefficients

bij =

234

25

54

5

84

5

339

100

18

539 24 73

20

3 9 48 3339

100

219

2024 1897

200

. (44)

We work in the MS scheme, and take the two-loop matching conditions accordingly[12]. As

an example, we use α = 1/127.9, sin2 θW = 0.2317, and αs = 0.116 at the scale MZ = 91.187

GeV. We also choose MSUSY = 1 TeV and use the top quark mass mt = 175 GeV. Note that

the value of αN is always close to that of αY since their one-loop beta fuctions are close in

value to each other and they are required to be unified at the scale MU .

Another possibility is to exploit the allowed variation of particle masses near the super-

string scale of MU ≈ 7gU · 1017GeV [13] in the MS scheme. Just as Yukawa couplings are

assumed to be subject only to the constraints of the unbroken gauge symmetry, the masses

of the superheavy 27 and 27∗ multiplet components may also be allowed to vary accordingly.

For example, take three copies of (u, d) + (u∗, d∗) and (νe, e) + (ν∗e , e

∗) with M ′ much below

12

MU , then between M ′ and MU ,

∆bY = 3×(

1

5+

3

5

)

=12

5, ∆b2 = 3× (3 + 1) = 12, (45)

∆b3 = 3× (2 + 0) = 6, ∆bN = 3×(

3

10+

2

5

)

=21

10. (46)

For M ′ ∼ 1016 GeV, gauge-coupling unification at MU ∼ 7 × 1017 GeV is again achieved.

We show in Fig. 4 the evolution of α−1i using also the two-loop coefficients

bij =

9 9 84

53

3 39 24 3

3 9 48 3

3 9 24 9

(47)

between MSUSY and M ′, and

bij =

253

25

81

520 102

25

26

5123 72 51

100

3 27 116 18

5

102

25

153

1024 957

100

(48)

betweenM ′ andMU . As an example, we use α = 1/127.9, sin2 θW = 0.2317, and αs = 0.123±

0.006 at the scale MZ = 91.187 GeV, MSUSY = 1 TeV and the top quark mass mt = 175

GeV. For αs(MZ) = 0.123, we find M ′ = 5.9 × 1016 GeV. It should be emphasized that

the sharp turn at M ′ should not be taken too literally but only as an indication that gauge

couplings may in fact evolve drastically near the unification energy scale. This possibility

allows us to have unification without having split multiplets containing both superheavy and

light components, as in most grand unified models. The size of αN is always very close to

that of αY since they have the same one-loop beta functions for scales beneath M ′ and are

required to be unified at MU . We note that the two-loop corrections are larger here than in

the MSSM due to the much larger particle content. We also observe that for αs(MZ) = 0.123

we obtain an MU which is about 1.5 times the superstring scale of 7gU · 1017GeV, whereas

the MU in most supersymmetric grand unified models is about 0.04 times that number.

13

7 Concluding Remarks

To accommodate a naturally light singlet neutrino, an extra U(1) factor is called for. It

has been shown[4] that the superstring-inspired E6 model is tailor-made for this purpose

as it contains U(1)N which has exactly the required properties. To obtain U(1)N as an

unbroken gauge group, we need to break E6 spontaneously along the N and N∗ directions

with superheavy 27’s and 27∗’s while preserving supersymmetry. This is impossible if the

superpotential is allowed only terms up to cubic order so that the theory is renormalizable.

On the other hand, the requirement of renormalizability may not be applicable at the super-

string unification scale, in which case the quartic term M−1 27 27∗ 27 27∗ in conjunction

with the quadratic term m 27 27∗ in the superpotential would result in 〈 27 〉 = 〈 27∗ 〉 =

(−2mM)1

2 without breaking supersymmetry.

The addition of U(1)N has several other interesting phenomenological consequences. (1)

The U(1)N neutral gauge boson Z’ mixes with the standard-model Z and affects the precision

data at LEP. From the present experimental error bars on the ǫ1,2,3 parameters, we find that

the U(1)N breaking scale could be as low as a few TeV. (2) The spontaneous breaking

of U(1)N is accomplished only with the presence of a mass term in the Higgs potential

which breaks the supersymmetry softly. Hence the reduced two-doublet Higgs potential at

the electroweak energy scale is not guaranteed to be that of the MSSM. In fact, the scalar

quartic couplings now depend also on a new Yukawa coupling f as well as the gauge coupling

gN . Assuming that gN = (5/3)1

2 g1, one result is that the upper bound on the lighter of the

two neutral scalar Higgs bosons is now 140 GeV instead of 128 GeV in the MSSM. (3)

The neutralino mass matrix also depends on f , hence there is a correlation here with the

Higgs sector. Such a connection is not present in the MSSM. (4) This model may also be

compatible with gauge-coupling unification. We identify two possible scenarios. One is just

like the MSSM with two light doublets presumably belonging to complete multiplets (of the

14

grand unified group) whose other members are superheavy; the other requires no light-heavy

splitting but assumes a large variation of superheavy masses near the unification scale.

ACKNOWLEDGEMENT

This work was supported in part by the U. S. Department of Energy under Grant No. DE-

FG03-94ER40837.

15

References

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[3] C. Athanassopoulos et al., Phys. Rev. Lett. 75, 2650 (1995).

[4] E. Ma, UCRHEP-T149, hep-ph/9507348 (1995).

[5] S. Nandi and U. Sarkar, Phys. Rev. Lett. 56, 564 (1986).

[6] G. Altarelli, R. Barbieri, and S. Jadach, Nucl. Phys. B369, 3(1992); B376, 444(E)

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[8] Particle Data Group, Phys. Rev. D50, 1173 (1994).

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[10] See for example J. F. Gunion and H. E. Haber, Nucl. Phys. B272, 1 (1986).

[11] See for example U. Amaldi, W. de Boer, and H. Furstenau, Phys. Lett. B260, 447

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ITP-838 (1992).

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Figure Captions

Fig. 1 : The tree-level Feynman diagrams due to the cubic couplings of the scalar√2Reχ to

the scalar Higgs doublets Φ1,2 which contribute to the quartic scalar couplings λ1,2,3,4

of the reduced Higgs potential given in Eq. (22).

Fig. 2 : The upper bound on the mass of the lighter of the two neutral scalar Higgs bosons

(mh)max for f = f0 and f = 0 as functions of cos2 β, as the maximum allowed values

of mh in the model discussed here and in the MSSM respectively. We have used

α = 1/127.9 and sin2 θW = 0.2317 at the MZ scale, mt = 175 GeV, and m = 1 TeV.

Fig. 3 : The two-loop evolution of the gauge couplings of the unification scenario involving three

complete 27 supermultiplets and one extra copy of (νe, e) and (Ec, N cE) with mass of

order MSUSY as explained in the text. We have used α = 1/127.9, sin2 θW = 0.2317,

and αs = 0.116 at the scale MZ = 91.187 GeV, MSUSY = 1 TeV, and mt = 175 GeV.

Fig. 4 : The two-loop evolution of the gauge couplings of the unification scenario explained

in the text which involves three complete 27 supermultiplets for scales above MSUSY

and with three additional copies of (u, d) + (u∗, d∗) and (νe, e) + (ν∗e , e

∗) with mass of

order the intermediate scale M ′. We have used α = 1/127.9, sin2 θW = 0.2317, and

αs = 0.123 ± 0.006 at the scale MZ = 91.187 GeV, MSUSY = 1 TeV, and mt = 175

GeV. The dashed lines correspond to αs(MZ) = 0.117 and 0.129.

17

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